In the spring of last year, Timothy F. Geithner wanted to leave his job. The Treasury secretary’s family was moving to New York for his son’s senior year in high school, and the commute to see them each weekend was sure to be arduous.

He ultimately failed to make the case, but it wasn’t for lack of trying. The effort illuminated one of the more intriguing relationships in Washington — the evolving rapport between President Obama and his high-wattage secretary of state.

Before he could depart, Geithner had to convince the White House. He needed to find someone who could glide through Senate confirmation, comprehend the growing debt crises, here and abroad, and communicate administration goals in last summer’s debt-ceiling debate to Congress, to Wall Street and to the world.

The talks gained traction inside the White House before being scuttled shortly before the debt-ceiling talks on Capitol Hill intensified, according to one former and two current administration officials.